$\DeclareMathOperator\End{End}\newcommand\Id{\mathrm{Id}}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SO{SO}$I start with some background, but people familiar with the subject may jump directly to the question.

Let $M^{4n}$ be a compact oriented smooth manifold. Recall that an *almost hypercomplex structure* on $M$ is a 3-dimensional sub-bundle $Q\subset \End(TM)$ spanned by three endomorphisms $I$, $J$ and $K$ satisfying the quaternionic identities: $I^2=J^2=-\Id$, $IJ=-JI=K$.

An *almost quaternionic structure* on $M$ is a 3-dimensional sub-bundle $Q\subset \End(TM)$ which is *locally* spanned by three endomorphisms with the above property.

In both cases one may assume (by an averaging procedure) that $M$ is endowed with a Riemannian metric $g$ compatible with $Q$ in the sense that $Q\subset \End^-(TM)$, i.e. $I$, $J$ and $K$ are almost Hermitian. Using this one sees that an almost hypercomplex or quaternionic structure corresponds to a reduction of the structure group of $M$ to $\Sp(n)$ or $\Sp(1)\Sp(n)$ respectively, but this is not relevant for the question below.

Notice that in dimension $4$ every manifold has an almost quaternionic structure (since $\Sp(1)\Sp(1)=\SO(4)$), but there are well-known obstructions to the existence of almost hypercomplex structures. For example $S^4$ is not even almost complex. Finally, here comes the question:

Are there any known topological obstructions to the existence of almost quaternionic structures on compact manifolds of dimension $4n$ for $n\ge 2$?

**EDIT:** Thomas Kragh has shown in his answer that there is no almost quaternionic structure on the sphere $S^{4n}$ for $n\ge 2$. I have found further obstructions in the literature and summarized them in my answer below.

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